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It is well known fact that it is very hard to prove Goldbach's strong conjecture but perhaps some weaker variations can be proved(or disproved) ,so my question is: Is it true that every even number greater than 10 can be represented as the sum of an odd prime number and an odd semiprime?

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@yunone,Thanks... – pedja Sep 13 '11 at 8:58
Have you seen Chen's theorem ( ? I don't know if that can be extended to just the prime + semiprime case, though – yatima2975 Sep 13 '11 at 9:01
@yatima2975: That's probably as good an answer as this question will get here. (I was going to post it myself before I rechecked the comments.) – anon Sep 13 '11 at 9:05
I wrote a small c++ program that verified the conjecture for all even numbers less than 600k and larger than 10. So perhaps it's true. – JSchlather Sep 13 '11 at 9:12
2 Has some references regarding improvements on Chen's theorem. – JSchlather Sep 13 '11 at 11:57

Some counterexamples: 12, 14, 16, 30. My perl program can't find any more smaller than 100000.

EDIT: I didn't know that semiprimes are defined to include squares. When I comment out the line that filters them, nothing is output up to 100000. I'll leave this answer here as an example of wrongness.

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9+3=12, 9+5=14, 9+7=16, 25+5=30. Are you forgetting that $p^2$ is semiprime in your code? – JSchlather Sep 13 '11 at 9:16
@Dan,12=3+9,9 is odd semiprime ;14=5+9 ;16=7+9; 30=5+25 ,25 is odd you are not right this time – pedja Sep 13 '11 at 9:18

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